For questions regarding harmonic functions.

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Critical points of a harmonic function

Suppose $\phi$ is harmonic on some compact, connected region of $\mathbb{R}^3$. Is there an algorithm that is guaranteed to find all critical points of $\phi$? (Obviously, these will all be saddle ...
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1answer
39 views

Find a harmonic conjugate for the function $u(x,y)=x^{3}+Axy^{2}$ if one exists - $u(x,y)$ might not be harmonic?

For $u(x,y) = x^{3} + Axy^{2}$, where $A$ is a real number, I need to find a harmonic conjugate, or if one does not exist, show that it does not exist. I began my approach to this problem by first ...
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34 views

Square integrable functions on the unit ball

In one dimension, the space $L^{2}([0,1], dx)$ of complex-valued square integrable functions on $[0,1]$ is well known to be separable, and sine and cosine provide an explicit Hilbert basis for it. My ...
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22 views

Find the maximum of the function $u(x,y)=3xy+2$

If $u$ is a solution of $\begin{cases} -\Delta u=0&\text{in}\ B(0,2)\\ u(x,y)=3xy+2& \text{for} (x,y)\in \partial B(0,2)\end{cases}$ then find maximum of $u$ in $\overline{B(0,2)}$ and the ...
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32 views

Find a harmonic function which goes to $0$ on the boundary, which is not identically $0$

Find a function which is harmonic on the area bounded by positive x axis and the line $y=x$, which goes to $0$ on the boundary, which is not identically $0$. Why doesn't it violate the max/min ...
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1answer
43 views

Intuition behind $\nabla \cdot \frac{1}{\rho}\nabla$?

Oftentimes instead of the Laplacian I notice the very similar operator $$\nabla \cdot \frac{1}{\rho}\nabla$$ What is the intuition behind this operator? How does it differ intuitively from the ...
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59 views

Fourier Transform to solve Laplace's equation in cylindrical coordinates

I am trying to solve $\nabla^2 u = 0$ in cylindrical polar coordinates (and radial symmetry) $$ \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{\partial^2 ...
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1answer
12 views

Laplace's equation 2 variable PDE/chain rule show function is a solution

The question is: 'Show that if f(x,y) is harmonic, then $f(x^2-y^2,2xy)$ is also harmonic using Laplace's equation: $\frac{\partial^2f}{\partial x^2} + \frac{\partial^2f}{\partial y^2} = 0 $. I end ...
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1answer
92 views

Find Green's function of quarter-plane with method of images

Consider a domain $$D= \{(x,y) : x>0, y>0 \}$$ Let $\textbf x = (x,y)$ and $\xi =(\xi_x , \xi_y)$. Find the Green's function, $G(\textbf x , \xi)$ such that $$\nabla ^2 G=\delta (\textbf x - ...
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1answer
22 views

Inequality involving harmonic functions over the ball and half ball

Let $B\subset \mathbb R^2$ be a unit ball. Let $B^+:=B\cap \{x_2\geq 0\}$ where we set $x=(x_1,x_2)\in \mathbb R^2$. Let $\omega\in C^1(\partial B)$ be given such that $|\nabla \omega|>0$ for all ...
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28 views

Verify that Poisson's integral formula is harmonic on the unit disk

Let $D = \{ z : |z| < 1\}$ be the unit disk and let $z \in D$ and $e^{i\theta} \in \partial D$. Let $h(e^{i\theta})$ be a function defined on $\partial D$ and let $\tilde h(z)$ be defined in ...
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1answer
35 views

Greens function for 2d laplace equation with neumann boundary conditions

I have a domain, $ D : {(x,y) : x>0 , y>0}$ Let $ \mathbf{x}= (x,y) $ and $\mathbf{\xi}= (\xi_x, \xi_y)$, The Greens function satisfying: $$\nabla^2G = \delta(\mathbf{x} - \mathbf{\xi} )$$ ...
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Is it posible to solve the Laplce equation on an open set with dirac delta boundary conditions?

By dirac deta boundary conditions, i mean there is an $y \in \partial D$ such that the value on that point is $\infty$ and 0 elsewhere. Intuitivelly i would think it should be true: the Laplace ...
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2answers
51 views

Show that $z^n+c$ is harmonic, $c\in\mathbb C$.

I would like to know how to show $$f(z) = z^n+c$$ for $c\in \mathbb C$ is harmonic over $D =\{|z|\leq r\}$. I know that if I express $z = x+iy$, then I can have $f=u+iv$, where $u$ and $v$ will be ...
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1answer
43 views

Intersections of the level curves of two (conjugate) harmonic functions

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ with $f(x,y)=e^{-x}(x\sin y-y\cos y)$. 1 Let $g$ be one of the conjugate harmonics of $f$ on $\mathbb{R}^2$ and assume the level curves of $f$ and $g$ ...
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1answer
23 views

What is the harmonic conjugate of $u=4xy-3x+5y$?

What is the harmonic conjugate of $u=4xy-3x+5y$? I got $u'x=4-y=v'y$ then I integrated $v'y$ to get $v= 2y^2-3y+h(x)$. Then I did $-u'y=v'x$ so, $5= h'(x)$ then I integrated $5$ with respect to $x$ ...
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1answer
26 views

Why is $\int_{B(0,\epsilon)}|\Phi(y)|dy$ bounded?

Why is $\int_{B(0,\epsilon)}|\Phi(y)|dy$ bounded ? If $\Phi$ is the fundamental solution of Laplace equation defined by $\Phi(x)=\begin{cases}-\frac1{2\pi}\log|x|& \text{if ...
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harmonic measure circle

I'm trying to compare the probability of a particle (performing Brownian Motion), starting a large distance away from a circle, passing through a specific section of that circle is approximately the ...
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1answer
48 views

Mean value of a subharmonic function, divided by the logarithm of radius, has a limit

I am pretty stuck on a homework problem on harmonic functions, or rather subharmonic functions (which for us are allowed to take the value $-\infty$). The statement is as follows: Supper $u$ is ...
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Deduce the definition of a harmonic function in the context of a Markov Chain

We know from PDE that a harmonic function $f$ satisfies the mean value property, namely, $f(x)$ = $\frac{1}{\vert{B_r(x)}\vert}\int_{B_r(x)}f(y)dy$ where $B_r(x)$ is the ball about $x$ with radius ...
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1answer
86 views

A harmonic function with sublinear growth at infinity is constant

Show that if $u$ is harmonic in $\mathbb R^n$ and $u=o(|x|)$, then $u$ is a constant.(Hint: use the solid version of the mean value property $u(x)=\frac{1}{\omega R^n} \int _{B_{R(x)}} u(y)dy$, and ...
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biharmonic complex analysis

Complex analysis texts typically discuss analytic functions whose real and imaginary components are harmonic and satisfy the Laplace equation, $\nabla^2 f = 0$. I am working with a complex function ...
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1answer
24 views

let $f(x)$ is real polynomial. Can we say that $f(\left| x \right|)$ is subharmonic?

A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le \frac{1}{{2\pi ...
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1answer
26 views

Is the product of two subharmonic function necessarily subharmonic?

Defin: A continuous function $\varphi :\mathbb{R} \to \mathbb{C}$ is subharmonic if and only if, for any closed disc in $U$ with centre $\lambda_0$ and radius $r$,$$\varphi ({\lambda _0}) \le ...
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18 views

Prove $\lim |u(x,y)|e^{-|x|}=0$ for harmonic function $u$

Let $\epsilon >0$ and $\Omega \subset \{(x,y): -\frac{\pi}{2}+\epsilon<y<\frac{\pi}{2}-\epsilon\}$. Let $u$ be harmonic function in $\Omega$ such that $u=0$ on $\partial \Omega$. Prove ...
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Summation of series involving $\sinh$ of a square root

In a physics problem that I assigned in one of my classes recently, the following series arises: $$ S = \sum_{\text{odd } n} \sum_{\text{odd } m} \frac{(-1)^{(n+m)/2}}{nm} \frac{\sinh( \pi \sqrt{n^2 + ...
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Square integrable harmonic function

Let $u$ be harmonic function on $\mathbb{R}^n$ such that $\int_{\mathbb{R}^n} u^2 dx1...dxn < +\infty$ (1). I've got to prove that it implies $u=0$. My idea was to prove that (1) implies that $u$ ...
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Co-efficient Problem For Univalent harmonic functions on Unit disk

The Clunie Sheil Small conjecture for the second co-efficient of a univalent harmonic function on the unit disk is as follows:- Suppose, $h(z)+\overline{g(z)}$ is a one-one harmonic function on the ...
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Generalized Coupon Collector's Problem [duplicate]

I just read about the coupon collector's problem where you're trying to find out how many coupons you need to collect on average before you get one complete set. This turns out to be $nH(n)$. I was ...
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Derive mean value property of harmonic function using this method

We can view harmonic function $u(x,y,z)$ as a wave function that does not depend on time $t$. Let $\overline{u}(r)$ be its average value on the circle with radius $r$. Assume we have proved that it ...
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1answer
29 views

Laplace equation for a lens-shaped volume - any approximate analytical solutions?

I need to solve Laplace equation for a lens (drop on a surface) with constant boundary condition on the top surface and zero on the bottom surface (this is the simplest case, not the only one I ...
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Is this a right calculation of Fourier transform?

I am trying to solve: Calculate Fourier transform (on $-\infty < x < \infty $) of: $$ f(x) = \begin{cases} 1, & -L<x<L \\ 0, & |x|\geq L \end{cases} $$ My ...
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Why does solving the Laplace equation on a graph using the method of relaxation get different results than a spectral method?

When I use the method of relaxation to solve Laplace's equation on a graph where the boundary conditions are fixing a set of nodes to either 0 or 1 then the solution ends up being entirely between 0 ...
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Surface Integral of a harmonic function and mean value property

I want to find a general expression of the following integral, where $h$ is a harmonic function (we're in $\mathbb{R}^2$): $\int_{|x-y|\leq a^2}\frac{h(y)}{\sqrt{a^2-|x-y|^2}}dy$ I think I can ...
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Harmonic function on 3-dimensional lamplighter group

Can anyone give an example of a non-constant bounded harmonic function on the Lamplighter group $\mathbb{Z}_2 \wr \ \mathbb{Z}^3$? This function should exist, because the random walk on this group has ...
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Fourier transform on forced linear harmonic oscillator

I'm having trouble solving the following equation: $\ddot x + \omega^2 x = a \sin \Omega t$, where $t >0$, $\omega \neq \Omega$, $x(0+) = 0 = \dot x(0+)$. It is asked to be done by Fourier ...
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30 views

Function $\phi (x,y) \mapsto \phi (u,v)$ [ie:…] under the transformation $f(z) = u(x,y) + v(x,y)$ where $f(z)$ is analytic & $d_z \,f(z)$ is not $0$

I am just having a bit of trouble understanding what I am being asked. The ie: in title says $\phi [x(u,v), y(u,v)]$ Part a) is asking to do the laplacian, $\nabla^2_{x,y} \phi (x,y)$ and to write ...
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Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that…

In reviewing complex analysis, I stumbled upon the following problem: Find a function $g(x,y)$ harmonic on $\{ 1<x^2+y^2<16\}$ such that $g(x,y)=3$ when $x^2+y^2=1$ and $g(x,y)=8$ when ...
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1answer
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Variational formulation of harmonicity on Riemannian manifolds

$\newcommand{\R}{\mathbb{R}}$ I am trying to follow a derivation of the first variation formula for the energy functional. (In "Selected Topics in Harmonic maps"). Here is the context: $M,N$ are ...
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1answer
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Isometries of Riemannian manifolds are harmonic?

Let $(M,g),(N,h)$ be two Riemannian manifolds. Assume $f:M \to N$ is an isometric immersion. Is $f$ harmonic? (i.e a critical point of the energy functional) (I know this is true when ...
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Is this function harmonic?

$$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to figure out if this function is harmonic. I worked out all the algebra and my conclusion is no because the numerator of the final fraction is $2x^3+2xy^2$ ...
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How to classify harmonic functions on the punctured disk without the Schwartz reflection principle?

I am working through old qual problems at the University of Minnesota and am trying to find an alternate solution to the following problem. Determine all continuous functions on $\{ z : 0 < \left| ...
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1answer
33 views

Partial derivative of x - is quotient rule necessary?

Let $$u(x,y)=\frac{x}{x^2+y^2}$$ I'm trying to determine if the given function is harmonic. I know that the 2nd partial derivative with respect to $x$ should, when added to the 2nd partial ...
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Find the harmonic conjugate of the following

I just want to make sure my reasoning is correct. I followed another similar question from this site. Also, is there a method I can use after coming to the conclusion below to check to make sure my ...
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35 views

How do I see if $g$ is a polynomial or not??

Let $u$ be a real valued harmonic function on $\mathbb{C}$. Let $g: \mathbb{R^2} \to \mathbb{R}$ be defined by: $$g(x,y)=\int_0^{2\pi}u(e^{i\theta}(x+iy))\sin \theta \,d\theta$$ Which of the ...
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Harmonic map into $S^n \times \mathbb{R}$

Consider a harmonic map $\Phi : \Sigma \to S^n \times \mathbb{R}$, where $\Sigma$ is a surface, and the metric on $S^n \times \mathbb{R}$ is given by the product metric. Choose local spherical (polar) ...
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1answer
38 views

A version of Casorati-Weierstrass for harmonic functions?

Suppose that $f:B(0,1)\setminus\left\{0\right\}\subset \mathbb{R}^n \to \mathbb{R}$ is a harmonic function. Clearly, the property that $\overline{f(B(0,\epsilon))}=\mathbb{R}$ for all $\epsilon>0$ ...
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52 views

Harmonic map and pullback metric

Let $\phi : M \to \mathbb{R}^n$ be a harmonic map, where $M$ is a Riemannian manifold. Let us take coordinates $(u_1, u_2,..., u_n)$ on $\mathbb{R}^n$ and express the Euclidean metric as $g = \Sigma ...
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2answers
25 views

Harmonic Motion - Fourier Approximation What does this mean below?

There is a method to solve systems under harmonic loading, harmonic balance method, which is basically obtaining fourier expansions of unknown response quantities and solving for coefficients of ...
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1answer
64 views

Interior estimate for derivatives of harmonic function

I'm learning PDE from the book of Trudinger and Gilbarg and I'm attempting to prove the following theorem: Let $\hspace{0.1ex}u\hspace{0.1ex}$ be harmonic in $\hspace{0.1ex}\varOmega \subset ...